Symmetry analysis for time-fractional convection-diffusion equation
aa r X i v : . [ n li n . S I] D ec Symmetry analysis for time-fractionalconvection-diffusion equation
Junjun Zhang a , Jun Zhang a,b, ∗ a Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China b Centre for Mathematics and Its Applications, The Australian National University, Canberra, ACT 0200,Australia
Abstract:
The time-fractional convection-diffusion equation is performed by Lie symmetry analysismethod which involves the Riemann-Liouville time-fractional derivative of the order α ∈ (0 , Keywords:
Fractional convection-diffusion equation, Riemann-Liouville fractional derivative, Lie sym-metry analysis, Group invariant solution
It is well known that Lie symmetry theory plays a significant role in the analysis of differentialequations[1-5]. The basic idea of this method is that the infinitesimal transformation leaves the set ofsolution manifold of the considered differential equation invariant. This efficient method invented by So-phus Lie is a highly algorithmic process, and it often involves lengthy symbolic computation. The methodsystematically unifies and extends well-known techniques to construct explicit solutions for differentialequations, especially for nonlinear differential equations. In recent years this method has a success-ful extension to discrete systems exhibiting solitons governed by nonlinear partial differential-differenceequations and pure difference equations[6-8].Recently the study of fractional differential equations(FDEs), as generalizations of classical integerorder differential equations, has attracted much attention due to an exact description of nonlinear phe-nomena in fluid mechanics, viscoelasticity, biology, physics, engineering and other areas of science[9-12].However, unlike the classical integer order derivatives, there exists a number of different definitions offractional order derivatives and corresponding FDEs. These definition differences lead to the FDEs hav-ing similar form but significantly different properties. It means that there exists no well-defined methodto analyze them systematically. As a consequence, several different analytical methods such as differen-tial transform method[13], Adomian decomposition method[14,15], invariant subspace method[16], Greenfunction approach[17] and symmetry analysis[18-26] have been formulated to reduce and solve FDEs.In this article, we consider the following time-fractional convection-diffusion equation ∂ α u∂t α = ( D ( u ) u x ) x + P ( u ) u x , < α < , (1)where ∂ α u∂t α is the Riemann-Liouville fractional derivative of order α with respect to the variable t . Thedefinition of this derivative is ∂ α u ( t, x ) ∂t α = ( ∂ n u∂t n , α = n, n − α ) ∂ n ∂t n R t ( t − s ) n − α − u ( s, x ) ds , ≤ n − < α < n, (2)here n ∈ N , ∂ n ∂t n is the usual partial derivative of integer order n with respect to t . *Email: [email protected] u t = ( D ( u ) u x ) x + P ( u ) u x , where u = u ( t, x ) , u t = ∂u∂t , u x = ∂u∂x , D ( u ) , P ( u ) are referred to the diffusivity and convective termsrespectively. The nonlinear convection-diffusion equation arises in many areas of science and engineeringsuch as being used to model the evolution of thermal waves in plasma[27].Our paper is organized as follows. In Section 2, we introduce Lie symmetry analysis for FDEs anddeduce the infinitesimals of symmetries for time-fractional convection-diffusion equation (1) by consideringthe different conditions for D ( u ) and P ( u ). In Section 3, the similarity reductions for Eq.(1) are presentedby Lie symmetries obtained in section 2. Some invariant solutions of Eq.(1) are given. Next we use Lie symmetry analysis for Eq.(1). Let us assume that Eq.(1) is invariant under thefollowing one parameter( ǫ ) continuous transformations x ∗ = x + ǫξ ( t, x, u ) + o ( ǫ ) ,t ∗ = t + ǫτ ( t, x, u ) + o ( ǫ ) ,u ∗ = u + ǫη ( t, x, u ) + o ( ǫ ) ,u ∗ x ∗ = u x + ǫη x + o ( ǫ ) ,u ∗ x ∗ x ∗ = u xx + ǫη xx + o ( ǫ ) ,∂ α u ∗ ∂t ∗ α = ∂ α u∂t α + ǫη tα + o ( ǫ ) , (3)where ξ, τ, η are infinitesimals and η x , η xx , η tα are extended infinitesimals of orders 1,2 and α respectively.The explicit expressions for η x , η xx are η x = D x η − u x ( D x ξ ) − u t ( D x τ ) ,η xx = D x ( η x ) − u xx ( D x ξ ) − u xt ( D x τ ) , where symbol D x stands the total derivative operator with respect to x , D x = ∂∂x + u x ∂∂u + u xx ∂∂u x + u tx ∂∂u t + · · · , with infinitesimal generator X = ξ ( t, x, u ) ∂∂x + τ ( t, x, u ) ∂∂t + η ( t, x, u ) ∂∂u . Since the lower limit of theintegral in Eq.(1) is fixed and, therefore it should be invariant with respect to the transformations(3).Such invariance condition arrives at τ ( t, x, u ) | t =0 = 0 . (4)The α th extended infinitesimal related to Riemann-Liouville fractional time derivative reads η tα = D αt η + ξ ( D αt u x ) − D αt ( ξu x ) + D αt ( uD t τ ) − D α +1 t ( τ u ) + τ ( D α +1 t u ) . (5)Here the symbol D t stands the total derivative operator with respect to t , i.e., D t = ∂∂t + u t ∂∂u + u xt ∂∂u x + u tt ∂∂u t + · · · , and the operator D αt is the total fractional derivative operator. By means of the generalized Leibnitzrule [12] D αt ( f ( t ) g ( t )) = ∞ X n =0 (cid:18) αn (cid:19) ( D α − nt f ( t ))( D nt g ( t )) , α > , where (cid:18) αn (cid:19) = ( − n − α Γ( n − α )Γ(1 − α )Γ( n + 1) , η tα = D αt η − α ( D t τ ) ∂ α u∂t α − ∞ X n =1 (cid:18) αn (cid:19) ( D nt ξ )( D α − nt u x ) − ∞ X n =1 (cid:18) αn + 1 (cid:19) ( D n +1 t τ )( D α − nt u ) . Furthermore, using the generalized chain rule for a compound function [28] d α u ( v ( t )) dt α = ∞ X n =0 n X k =0 (cid:18) nk (cid:19) ( − v ( t )) k n ! ∂ α ( v n − k ( t )) ∂t α d n u ( v ( t )) dv n along with the above generalized Leibnitz rule with f ( t ) = 1, the first term D αt η in η tα can be written as D αt η = ∂ α η∂t α + η u ∂ α u∂t α − u ∂ α η u ∂t α + ∞ X n =1 (cid:18) αn (cid:19) ∂ n η u ∂t n D α − nt ( u ) + µ, where µ = ∞ X n =2 n X m =2 m X k =2 k − X r =0 (cid:18) αn (cid:19)(cid:18) nm (cid:19)(cid:18) kr (cid:19) t n − α Γ( n + 1 − α ) ( − u ) r k ! ∂ m u k − r ∂t m ∂ n − m + k η∂t n − m + k ∂u k . Therefore η tα = ∂ α η∂t α + ( η u − αD t τ ) ∂ α u∂t α − u ∂ α η u ∂t α + µ + ∞ X n =1 (cid:2)(cid:18) αn (cid:19) ∂ n η u ∂t n − (cid:18) αn + 1 (cid:19) D n +1 t ( τ ) (cid:3) D α − nt u − ∞ X n =1 (cid:18) αn (cid:19) ( D nt ξ )( D α − nt u x ) . For the invariance of Eq.(1) under transformations(3), we have ∂ α u ∗ ∂t ∗ α = ( D ( u ∗ ) u ∗ x ∗ ) x ∗ + P ( u ∗ ) u ∗ x ∗ , < α < u = u ( t, x ) of Eq.(1). Expanding Eq.(6) about ǫ = 0, making use of infinitesimals andtheir extensions, equating the coefficients of ǫ , and neglecting the terms of higher power of ǫ , we obtainthe following invariant equation of Eq.(1)[ η tα − ( P ′ ( u ) u x + D ′′ ( u )( u x ) + D ′ ( u ) u xx ) η − ( P ( u ) + 2 u x D ′ ( u )) η x − D ( u ) η xx ] | Eq. (1) = 0 . (7)Here we assume that D ( u ) and P ( u ) are not equal to zero, otherwise Eq.(1) would be another equationthat have been considered in [20]. Substituting the expressions for η tα , η x and η xx into the above equationand equating various powers of derivatives of u to zero, we obtain an over determined system of linearequations. They are ξ t = ξ u = τ x = τ u = η uu = 0 ,P ( u )( ξ x − ατ t ) − P ′ ( u ) η − D ′ ( u ) η x − D ( u )(2 η xu − ξ xx ) = 0 ,D ′′ ( u ) η + D ′ ( u )( η u − ξ x + ατ t ) = 0 ,D ( u )(2 ξ x − ατ t ) − D ′ ( u ) η = 0 , (8) ∂ α η∂t α − u ∂ α η u ∂t α − P ( u ) η x − D ( u ) η xx = 0 , (cid:18) αn (cid:19) ∂ n η u ∂t n − (cid:18) αn + 1 (cid:19) D n +1 t ( τ ) = 0 , n = 1 , , · · · . In order to solve the above system, we consider the following different conditions for D ( u ), P ( u ) andobtain their corresponding Lie symmetries of Eq.(1). And if α = 1, Eq.(1) becomes partial differentialequation which has been considered by Oron, Rosenau[29] and Edwards[30]. Therefore, α ∈ (0 ,
2) and α = 1 in our paper. Case 1 D ( u ) and P ( u ) arbitrarySolving the determining equations (8), we obtain the explicit form of infinitesimals ξ = a , τ = 0 , η = 0 , a is an arbitrary constant. Hence the infinitesimal generator is X = a ∂∂x . (9) Case 2 D ( u ) = u k ( k = 0), P ( u ) = β ( β = ± ξ = a + a x, τ = a tα , η = a uk , where a and a are arbitrary constants. Hence in this case Eq.(1) admits a two-parameter group withinfinitesimal generators X = ∂∂x , X = x ∂∂x + tα ∂∂t + uk ∂∂u , (10)which are the basis of 2-dimensional Lie algebras admitted by Eq.(1).The other cases are listed in Table 1.No. D ( u ) P ( u ) ξ τ η u k ( k = 0 , − , α − α ) βu k ( β = ± a a t − a αuk u k ( k = 0) βu γ ( β = ± , γ = k ) a + a x γ − kα ( γ − k ) a t − a uγ − k u − βu − ( β = ± a + a e − βx a t a αu + a βue − βx u α − α βu α − α ( β = ± a a t + a t a ( α − u + a ( α − tu β ( β = ± a a u + h ( t, x ) , where h ( t, x ) satisfies ∂ α h ( t,x ) ∂t α = βh x + h xx βu γ ( β = ± , γ = 0) a + a x a α t − a γ u Table 1: Infinitesimals of Eq.(1)In the above table, a , a , a are three arbitrary parameters. Like case 1 and case 2, the infinitesimalgenerators of Eq.(1) in different cases can easily obtained. Next we will use infinitesimal generators todeduce the similarity reductions and construct invariant solutions of Eq.(1). The definition of groupinvariant solution of FDEs has given in [22]. Here we use it directly. Case 1. D ( u ) arbitrary, P ( u ) arbitraryThe infinitesimal generator is X = ∂∂x . The characteristic equations become dx dt du , which have two invariants t, u. Thus, the similarity transformation is u = ϕ ( t ) . (11)Substitution of (11) into Eq.(1) leads to ϕ ( t ) satisfying the reduced fractional ordinary differential equa-tion d α ϕ ( t ) dt α = 0 . (12)Hence,the group invariant solutions of Eq.(1) are given by u = (cid:26) c t α − , < α < ,c t α − + c t α − , < α < . where c , c are arbitrary constants. 4 ase 2. D ( u ) = u k ( k = 0 , − , α − α ), P ( u ) = β ( β = ± X = x ∂∂x + tα ∂∂t + uk ∂∂u is dxx = αdtt = kduu . Solving the above equation, we get the similarity transformation u = t αk ϕ ( ζ ) , ζ = xt − α . (13)Substituting transformation(13) into Eq.(1) leads to ∂ α ( t αk ϕ ( ζ )) ∂t α = t αk − α ( ϕ k d ϕdζ + kϕ k − ( dϕdζ ) + β dϕdζ ) . (14)Because α ∈ (0 ,
2) and α = 1, according to the definition of the Riemann-Liouville fractional derivative,we should consider 0 < α < < α < < α <
1, for the similarity transformation(13) becomes ∂ α ( t αk ϕ ( ζ )) ∂t α = 1Γ(1 − α ) ∂∂t Z t ( t − s ) − α s αk ϕ ( xs − α ) ds. (15)Let θ = ts , then Eq.(15) can be written as ∂ α ( t αk ϕ ( ζ )) ∂t α = 1Γ(1 − α ) ∂∂t Z ∞ ( t − tθ ) − α ( tθ ) αk ϕ ( ζθ α ) tθ dθ = ∂∂t [ t αk − α +1 − α ) Z ∞ ( θ − − α θ α − αk − ϕ ( ζθ α ) dθ ]= ∂∂t [ t αk − α +1 ( K αk , − α α ϕ )( ζ )]= t αk − α (1 + αk − α − αζ ddζ )[( K αk , − α α ϕ )( ζ )]= t αk − α [( P αk − α,α α ( ϕ ))( ζ )] , where P τ,αβ is Erdelyi-Kober fractional derivative operator and its definition is( P τ,αβ ϕ )( ζ ) = n − Y j =0 ( τ + j − β ζ ddζ )[ K τ + α,n − αβ ( ϕ )( ζ )] , ζ > , β > , α > , n = (cid:26) [ α ] + 1 , α / ∈ N,α , α ∈ N. here ( K τ,αβ ϕ )( ζ ) = ( α ) R ∞ ( θ − α − θ − ( τ + α ) ϕ ( ζθ β ) dθ , α > ,ϕ ( ζ ) , α = 0 . When 1 < α <
2, we can also obtain ∂ α ( t αk ϕ ( ζ )) ∂t α = t αk − α [( P αk − α,α α ( ϕ ))( ζ )] , by using the same method. Then Eq.(1) can be reduced into an ordinary differential equation of fractionalorder ( P αk − α,α α ϕ )( ζ ) = ϕ k d ϕdζ + kϕ k − ( dϕdζ ) + β dϕdζ . (16)As for other cases, Eq.(1) can also be reduced by the similarity transformations corresponding to otherinfinitesimal generators. The results are as follows. Case 3. D ( u ) = u k ( k = 0), P ( u ) = βu k ( β = ± u = t − αk ψ ( ζ ) along with the similarity variable ζ = x reduces Eq.(1) tothe nonlinear ordinary differential equation of the form5 ψdζ + kψ − ( dψdζ ) + β dψdζ − Γ(1 − αk )Γ(1 − αk − α ) ψ − k ( ζ ) = 0 , (17)which is corresponding to the infinitesimal generator t ∂∂t − αuk ∂∂u . Case 4. D ( u ) = u k ( k = 0), P ( u ) = βu γ ( β = ± , γ = k )The similarity transformation u = x k − γ H ( ω ) along with the similarity variable ω = tx − b , b = γ − kα ( γ − k ) reduces Eq.(1) to the nonlinear ordinary differential equation of fractional order of the form( P − α,α − H )( ω ) = ω α [ ( γ +1)( k − γ ) H k +1 − bω ( k + γ +2 k − γ − b ) H k dHdω + b ω ( kH k − ( dHdω ) + H k d Hdω ) + βk − γ H γ +1 − bβωH γ dHdω ] , (18)which is corresponding to the infinitesimal generator x ∂∂x + γ − kα ( γ − k ) t ∂∂t + uk − γ ∂∂u . Case 5. D ( u ) = u − , P ( u ) = βu − ( β = ± u = e βx G ( ζ ) along with the similarity variable ζ = t reduces Eq.(1) tothe nonlinear ordinary differential equation of fractional order of the form d α G ( ζ ) dζ α = 0 , (19)which is corresponding to the infinitesimal generator e − βx ( ∂∂x + βu ∂∂u ).Because the solutions of Eq.(19) is G ( ζ ) = (cid:26) c ζ α − , < α < ,c ζ α − + c ζ α − , < α < . where c , c are arbitrary constants, the group invariant solution of Eq.(1) is u = (cid:26) c t α − e βx , < α < , ( c t α − + c t α − ) e βx , < α < . Case 6. D ( u ) = u α − α , P ( u ) = βu α − α ( β = ± u = t α − F ( ζ ) along with the similarity variable ζ = x reduces Eq.(1)to the nonlinear ordinary differential equation of the form d Fdζ + 2 α − α F − ( dFdζ ) + β dFdζ = 0 . (20)which is corresponding to the infinitesimal generator t ∂∂t + ( α − tu ∂∂u . Case 7. D ( u ) = 1 and P ( u ) = β ( β = ± u = e x Q ( ζ ) along with the similarity variable ζ = t reduces Eq.(1) tothe nonlinear ordinary differential equation of fractional order of the form d α Q ( ζ ) dζ α = (1 + β ) Q ( ζ ) , (21)which is corresponding to the infinitesimal generator ∂∂x + u ∂∂u . Because the solution of Eq.(21) is Q ( ζ ) = ζ α − E α,α [(1 + β ) ζ α ] , the group invariant solution of Eq.(1) is u = t α − e x E α,α [(1 + β ) t α ] , where E α,β ( z ) is the Mittag-Leffler function[11]. Case 8. D ( u ) = 1 and P ( u ) = βu γ ( β = ± , γ = 0)The similarity transformation u = t − α γ φ ( σ ) along with the similarity variable σ = xt − α reducesEq.(1) to the nonlinear ordinary differential equation of fractional order of the form( P − α γ − α,α α φ )( σ ) = d φdσ + βφ γ dφdσ , (22)which is corresponding to the infinitesimal generator x ∂∂x + tα ∂∂t − uγ ∂∂u .6 Summary and discussion
In this paper, we illustrate the application of Lie symmetry analysis to study time-fractional convection-diffusion equation. We consider the group classification of this equation for two variable functions. Eightcases are discussed. In every case Lie point symmetries are derived and similarity reductions of thisequation are performed by means of non-trivial Lie point symmetry. In some cases, the time fractionalconvection-diffusion equation can be transformed into a nonlinear ODE of fractional order. In other cases,this equation can be reduced to a nonlinear ODE. Some invariant solutions are given in some cases. In ad-dition, it is easily shown that the infinitesimal generator admitted by time-fractional convection-diffusionequation in each cases can form Lie algebra and the dimension of Lie algebra is decided by the number ofparameters in transformations. It is necessary to remark that case 7 is different from other cases becausein that case Lie algebra is infinite dimensional and in other cases Lie algebra is finite dimensional.Lie symmetry analysis also can be used for other time-fractional differential equations. But thereare little conclusions on the symmetry property for one kind of fractional equations. This is a possibledirection for future work.
Acknowledgements
This work is supported by natural science foundation of Zhejiang Province (Grant No.Y6100611) andthe national natural science foundation of China(Grant No.11371323)